To eliminate the arbitrary constant $a$ from the equation $u = ax^2 + a^2y + z$, we can follow these steps:

Step 1: Find the partial derivatives of $u$ with respect to $x$, $y$, and $z$.

1. Partial derivative with respect to $x$: $u_x = \frac{\partial}{\partial x}(ax^2 + a^2y + z) = 2ax$

2. Partial derivative with respect to $y$: $u_y = \frac{\partial}{\partial y}(ax^2 + a^2y + z) = a^2$

3. Partial derivative with respect to $z$: $u_z = \frac{\partial}{\partial z}(ax^2 + a^2y + z) = 1$

Step 2: Solve for $a$ in terms of the partial derivatives.

From $u_y = a^2$, we have: $a = \pm \sqrt{u_y}$ Substitute this into the expression for $u_x$: $u_x = 2a x = 2 (\pm \sqrt{u_y}) x$ Thus: $u_x = \pm 2x \sqrt{u_y}$ Now we have a relationship between the derivatives $u_x$ and $u_y$, and the constant $a$ has been eliminated.

Step 3: Form the first partial differential equation.

The first partial differential equation is: $u_x = 2x \sqrt{u_y}$ This is the equation that eliminates $a$ and relates $u_x$ and $u_y$.

Step 4: Derive a second equation.

Next, to find a second PDE, we can consider another relation. For example, substitute $a^2 = u_y$ into the original equation and differentiate again. However, for simplicity, we'll eliminate $a$ in another way using the relationship between the partial derivatives.

Taking the second derivative of $u$ with respect to $y$ will provide a second equation. Consider: $u_{yy} = \frac{\partial}{\partial y}(a^2) = 2a \cdot 2y = 2a$

This will give a second relationship once you substitute $u_x$ into the equation.

If you'd like further clarification or more details on forming the second equation, feel free to ask!

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Do you want to dive deeper into the second equation or have more examples? Here are some additional questions:

1. How would you solve for $a$ in terms of the other variables if the equation were different?
2. Can you use these methods to derive more complicated PDEs?
3. How can this process be applied to higher order PDEs?
4. What happens if we take additional derivatives of $u$?
5. How would you approach solving this system of PDEs?

Tip: When eliminating arbitrary constants, take advantage of partial derivatives, as they often simplify expressions and eliminate dependencies on arbitrary parameters.